Consider the following graphs:

A$.$ Are these graphs strongly connected? Explain why or why not.

B$.$ Are these graphs aperiodic? If a graph is periodic, compute its period.

Consider the basic averaging opinion dynamics model $($in timestep t$,$ each

node adopts the time $t-1$ average opinion of its out$-$neighbors$).$

A$.$ Give an example of a strongly connected, periodic graph with a period of $4.$

B$.$ In your graph from $2($A$),$ what happens to the nodes' initial opinions over time?

A qualitative answer is fine here.

C$.$ Give an example of a graph with at least $4$ nodes in which the averaging

opinion dynamics converges to a state in which not all nodes have the same

opinion.

D$.$ Give an example of a graph with at least $4$ nodes in which the averaging

opinion dynamics converges to consensus $($every node has the same opinion$),$

and the common opinion is exactly the average of all nodes' initial opinions.